Available at Applications and Applied http://pvamu.edu/pages/398/asp Mathematics
Vol. 1, No. 2 (2006) pp. 125 – 139 (AAM): ISSN: 1932-9466 An International Journal
GLOBAL STABILITY RESULTS AND WELL POSEDNESS OF AN SI AGE-STRUCTURED EPIDEMIC MODEL WITH VERTICAL TRANSMISSION
M. EL-DOMA Center for Advanced Mathematical Sciences (CAMS) College Hall, Room 426 American University of Beirut P. O. Box: 11-0236 Beirut-Lebanon E-mail: [email protected] Telephone: (+961) 1 374444 or 374374 ext. 4393 Fax: (+961) 1 365087
Abstract An SI age-structured epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility and mortality rates depend on age and the force of infection of proportionate mixing assumption type. We prove the well posedness of the model as well as the global stability for endemic equilibriums. Keywords: Vertical transmission; Horizontal transmission; Age-structure; Epidemic; Global stability; Proportionate mixing; Well posedness. MSC 2000: 45K05; 45M10; 35A05; 35B30; 35B35; 35B45; 35L40; 92D30; 92D25.
1 Introduction Several recent papers and books have dealt with SI age-structured epidemic models, for example, Busenberg, et al. (1993), (1991), El-Doma (2006), (2004a), (2004b), (2004c),
125 126 Mohammed El-Doma
(2000), (1987), Brauer (2002a), (2002b), Louie, et al. (1994), Anderson, et al. (1991), May, et al. (1988) and Hoppenstead (1975). In this paper, we study an SI age-structured epidemic model, where age is the chrono- logical age i.e., the elapse of time since birth. The disease causes so few fatalities that they can be neglected and is horizontally as well as vertically transmitted. Horizon- tal transmission of infection is the transfer of infection through some direct or indirect contact with infected individuals, for example, malaria is horizontally transmitted. A particular form of horizontal transmission known as proportionate mixing is assumed in this paper. Vertical transmission of infection is the passing of infection to offspring of infected parentage, for example, AIDS, chagas and hepatitis B are vertically (as well as horizontally) transmitted diseases. This form of transmission plays an important role in maintaining some diseases, for example, see Busenberg, et al. (1993). We note that some sexually transmitted diseases such as herpes and AIDS, are of SI type. We establish the well posedness of the model and prove global stability results and show that, if q =6 1, then the endemic equilibrium is globally stable. We also show that if q = 1, and other conditions hold, then the endemic equilibrium is also globally stable. The local stability of the endemic equilibrium as well as the global stability of the disease-free equilibrium are reported in El-Doma (2004a). We note that if q = 1, then either there exists a unique endemic equilibrium, which is globally stable, or problem (2.1) gives rise to a continuum of endemic equilibriums, in the case of non-fertile infectibles. The global stability results that we obtain are under very general conditions, and, in fact, we do not require any condition other than the existence of a unique endemic equilibrium. The organization of this paper is as follows: in section 2 we describe the model and obtain the model equations; in section 3 we reduce the model equations to several sub- systems and prove the well posedness of the model equations; in section 4 we determine the steady states; in section 5 we prove global stability results; in section 6 we conclude our results.
2 The Model We consider an age-structured population of variable size exposed to a communicable disease. The disease is vertically as well as horizontally transmitted and causes so few fatalities that they can be neglected. We assume the following. 1. s(a, t) and i(a, t), respectively, denote the age-density for susceptible and infective of age a at time t. Then a Z 2 s(a, t)da = total number of susceptible at time t of ages between a1 and a2, a 1a Z 2 i(a, t)da = total number of infective at time t of ages between a1 and a2. We a1 assume that the total population consists entirely of susceptible and infective. 2. Let k(a, a0) denote the probability that a susceptible individual of age a is infected 0 0 0 by an infective of age a . We further assume that, k(a, a )=k1(a).k2(a ), which is known as the “proportionate mixing assumption”, see Dietz, et al. (1985). There- fore the horizontal transmission of the disease occurs at the following rate: ∞ Z 0 0 0 k1(a)s(a, t) k2(a )i(a ,t)da , 0 AAM: Intern. J., Vol. 1, No. 2 (2006) 127
where k1(a) and k2(a) are bounded, nonnegative, continuous functions of a. The term ∞ Z 0 0 0 k1(a) k2(a )i(a ,t)da , 0 is called “force of infection” and we let ∞ Z λ(t)= k2(a)i(a, t)da. 0 3. The fertility rate β(a) is a nonnegative, continuous function, with compact support [0,A], (A>0). The number of births of susceptible per unit time is given by ∞ s(0,t)=Z β(a)[s(a, t)+(1− q)i(a, t)]da, q ∈ [0, 1], where q is the probability 0 of vertically transmitting the disease. Accordingly, all newborns from susceptible parents are susceptible but a portion q of newborns from infected parents are in- fective, i.e., they acquire the disease via birth (vertical transmission) and therefore, ∞ i(0,t)=qZ β(a)i(a, t)da. 0 4. The death rate, µ(a), is the same for susceptible and infective and µ(a) is a non- negative, continuous function and ∃ a0 ∈ [0, ∞) such that µ(a) > µ>¯ 0, ∀ a>a0 and µ(a2) >µ(a1), ∀ a2 >a1 >a0.
5. The initial age distributions s(a, 0) = s0(a) and i(a, 0) = i0(a) are continuous, nonnegative and integrable functions of a ∈ [0, ∞). These assumptions lead to the following system of nonlinear integro-partial differen- tial equations with non-local boundary conditions, which describes the dynamics of the transmission of the disease.
∂s(a, t) ∂s(a, t) + + µ(a)s(a, t)=−k1(a)s(a, t)λ(t),a>0,t>0, ∂a ∂t ∂i(a, t) ∂i(a, t) + + µ(a)i(a, t)=k1(a)s(a, t)λ(t),a>0,t>0, ∂a ∂t ∞ s(0,t)=Z β(a)[s(a, t)+(1− q)i(a, t)]da, t ≥ 0, 0 (2.1) ∞ i(0,t)=q Z β(a)i(a, t)da, t ≥ 0, 0 ∞ Z λ(t)= k2(a)i(a, t)da, t ≥ 0, 0 s(a, 0) = s0(a),i(a, 0) = i0(a),a≥ 0. We note that problem (2.1) is an SI age-structured epidemic model that has been studied in El-Doma (2004a), where the steady states are determined and the local asymp- totic stability of the endemic equilibrium and the disease-free equilibrium as well as the 128 Mohammed El-Doma
the global stability of the disease-free equilibrium are reported. The same model but with k1 and k2 constants is studied in El-Doma (1987). Also, in Brauer (2002a), (2002b), problem (2.1) is studied when k1 and k2 are constants and q = 0 (the case of no vertical transmission). In what follows, we establish the well posedness of the model equations and prove the global stability of the endemic equilibrium when q =16 . We also show that if q = 1 and another condition holds, then the endemic equilibrium is globally stable.
3 Reduction of the Model and its well posedness In this section, we develop some preliminary formal analysis of problem (2.1) and show that problem (2.1) is well posed. We define p(a, t)byp(a, t)=s(a, t)+i(a, t). Then from (2.1), by adding the equations, we find that p(a, t) satisfies the following McKendrick-Von Forester equation: ∂p(a, t) ∂p(a, t) + + µ(a)p(a, t)=0,a>0,t>0, ∂a ∂t ∞ (3.1) p(0,t)=B(t)=Z β(a)p(a, t)da, t ≥ 0, 0 p(a, 0) = p0(a)=s0(a)+i0(a),a≥ 0. Note that problem (3.1) has a unique solution that exists for all time, see Bellman, et al. (1963), Feller (1941) and Hoppensteadt (1975). The unique solution is given by p (a − t)π(a)/π(a − t), a>t, p(a, t)= 0 (3.2) B(t − a)π(a), a ∗ B(t)=[c + θ(t)]ep t, (3.3) where p∗ is the unique real number which satisfies the following characteristic equation: ∞ ∗ Z β(a)π(a)ep tda =1, (3.4) 0 θ(t) is a function such that θ(t) → 0ast →∞and c is a constant. Also, from (2.1), s(a, t) and i(a, t) satisfy the following systems of equations: ∂s(a, t) ∂s(a, t) + + µ(a)s(a, t)=−k (a)s(a, t)λ(t),a>0,t>0, 1 ∂a ∂t ∞ (3.5) s(0,t)=Z β(a)[s(a, t)+(1− q)i(a, t)]da, t ≥ 0, 0 s(a, 0) = s0(a),a≥ 0, AAM: Intern. J., Vol. 1, No. 2 (2006) 129 ∂i(a, t) ∂i(a, t) + + µ(a)i(a, t)=k (a)s(a, t)λ(t),a>0,t>0, 1 ∂a ∂t ∞ (3.6) i(0,t)=q Z β(a)i(a, t)da, t ≥ 0, 0 i(a, 0) = i0(a),a≥ 0. Using (3.1)-(3.2), we obtain that B(t) satisfies ∞ ∞ Z Z π(a) B(t)= β(a)π(a)B(t − a)da + β(a)p0(a − t) da. (3.7) 0 t π(a − t) Using (3.7) and Gronwall0s inequality, we obtain (k β(a)k∞−µ∗)t B(t) ≤kβ(a)k∞ kp0(a)kL1[0,∞) e , (3.8) where µ∗ is given by µ∗ = inf µ(a). (3.9) a∈[0,∞) From (3.2) and (3.8), we obtain the following a priori estimate: ∞ Z (k β(a)k∞−µ∗)t p(a, t)da ≤kp0(a)kL1[0,∞) e . (3.10) 0 By integrating problem (3.6) along characteristic lines t − a = const., and using s(a, t)= p(a, t) − i(a, t), we find that i(a, t) satisfies t − R [µ(a−t+τ)+k1(a−t+τ)λ(τ)]dτ i0(a − t)e 0 p (a − t)π(a) t 0 − R k1(a−t+τ)λ(τ)dτ + h1 − e 0 i ,a>t, π(a − t) i(a, t)= (3.11) a −R k1(τ)λ(t−a+τ)dτ i(0,t− a)π(a)e 0 − t k (τ)λ(t−a+τ)dτ − h − R0 1 i +B(t a)π(a) 1 e ,a n 1 o E = i(a, t):i(., t) ∈ L ([0, ∞)); C[0,t0]),a∈ [0, ∞),t∈ [0,t0], ki(a, t)k = sup ki(a, t)kL1 , t∈[0,t0] 1 where C[0,t0] denotes the Banach space of continuous functions in [0,t0] and L ([0, ∞)) denotes the space of equivalent classes of Lebesgue integrable functions. We note that E is a Banach space. In order to facilitate our future calculations, we need the following lemma: 130 Mohammed El-Doma Lemma 3.1 Suppose that x, y ≥ 0, then |e−y − e−x|≤|y − x| . Proof. Let f(x)=e−x, then use the mean value theorem to establish the required result. In the next theorem, we prove the existence and uniqueness of solution to problem (2.1) via a fixed-point theorem. Theorem 3.2 Problem (2.1) has a unique solution that exists for all time. Proof. Define the set Q by Q = {i(a, t) ∈ E,i(a, t) ≥ 0, ki(a, t)k≤M} , where M is a constant which satisfies the following: (kβ(a)k∞−µ∗)t0 M>kp0(a)kL1 e . (3.12) We note that Q is a closed set in E. Now, for fixed initial age-distributions s0(a),i0(a), and p0(a), define the mapping T : Q ⊂ E → E by t − R [µ(a−t+τ)+k1(a−t+τ)λ(τ)]dτ i0(a − t)e 0 p (a − t)π(a) t 0 − R k1(a−t+τ)λ(τ)dτ + h1 − e 0 i ,a>t, π(a − t) Ti(a, t)= (3.13) a − R k1(τ)λ(t−a+τ)dτ i(0,t− a)π(a)e 0 − t k (τ)λ(t−a+τ)dτ − h − R0 1 i +B(t a)π(a) 1 e ,a We note that since s0(a),i0(a), and p0(a), and B(t) are non-negative, continuous and integrable functions, we can use (3.8) and (3.10) to show that T maps Q into Q. Now, we look for a fixed point of this mapping to provide existence and uniqueness of solution for problem (2.1). To this end, we let i(a, t) and i1(a, t) be elements of Q, then using (3.8)-(3.10), and Lemma (3.1), we obtain the following: t Z kTi(., t) − Ti1(., )kL1 ≤ K(M,t0) ki(., σ) − i1(., σ)kL1 dσ, (3.14) 0 where K(M,t0) is a constant which depends on M and t0. Therefore, kTi(., t) − Ti1(., t)k≤t0K(M,t0) ki(., σ) − i1(., σ)k . (3.15) And thus, by induction, for each positive integer n, we obtain [t K(M,t )]n kT ni(., t) − T ni (., t)k≤ 0 0 ki(., t) − i (., t)k . (3.16) 1 n! 1 Inequality (3.16) implies that there exists a positive integer N such that T N is a strict contraction on Q. Thus T has a unique fixed point in Q. Since t0 is arbitrary, it follows that problem (2.1) has a unique solution that exists for all time. This completes the proof of the theorem. In the next theorem, we show that solutions of problem (2.1) depend continuously on the initial age-distributions, therefore, problem (2.1) is well posed. AAM: Intern. J., Vol. 1, No. 2 (2006) 131 Theorem 3.3 Let p(a, t) and p1(a, t) be two solutions of problem (2.1) corresponding to initial age-distributions p0(a),s0(a),i0(a) and p01(a),s01(a),i01(a), respectively. Also, suppose that p(0,t)=B(t) and p1(0,t)=B1(t), and let i(a, t) and i1(a, t) be the corre- sponding solutions of problem (3.6). Then the following properties hold: (kβ(a)k∞−µ∗)t |B(t) − B1(t)|≤kβ(a)k∞ kp0(a) − p01(a)kL1 e , (3.17) (kβ(a)k∞−µ∗)t kp(., t) − p1(., t)kL1 ≤kp0(a) − p01(a)kL1 e , (3.18) ki(., t) − i1(., t)kL1 ≤ k k h β ∞t0 i tK(M,t0) ki0(a) − i01(a)kL1 +2kp0(a) − p01(a)kL1 e e . (3.19) Proof. Note that (3.17) and (3.18) follow directly from (3.8) and (3.10), respectively, by linearity. To obtain (3.19), first we use (3.11) and (3.17)-(3.18), and then (3.14) to obtain the following: k k h β ∞t0 i ki(., t) − i1(., t)kL1 ≤ ki0(a) − i01(a)kL1 +2kp0(a) − p01(a)kL1 e t Z + K(M,t0) ki(., σ) − i1(., σ)kL1 dσ. 0 Now, the foregoing inequality yields (3.19) by the aid of Gronwall0s inequality. This completes the proof of the theorem. We note that (3.17)-(3.19), show that solutions of problem (2.1) depend continuously on the initial age-distributions, and therefore, problem (2.1) is well posed. 4 The Steady States In this section, we look at the steady state solution of problem (2.1), under the assumption that the total population has already reached its steady state distribution p∞(a)=cπ(a), i.e., we assume that (3.4) is satisfied with p∗ =0, see, for example, Busenberg, et al. (1988). A steady state s∗(a), i∗(a) and λ∗ must satisfy the following equations: ds∗(a) + µ(a)s∗(a)=−k (a)s∗(a)λ∗,a>0, 1 da (4.1) ∗ − ∗ s (0) = c i (0), di∗(a) + µ(a)i∗(a)=k (a)s∗(a)λ∗,a>0, 1 da (4.2) ∞ ∗ Z ∗ i (0) = q β(a)i (a)da, 0 132 Mohammed El-Doma ∞ ∗ Z ∗ λ = k2(a)i (a)da. (4.3) 0 Anticipating our future needs, we define a threshold parameter R0, and is given by ∞ a ∞ Z Z Z ∞ a cq β(a)π(a)k1(σ)dσda k2(a)π(a)da Z Z 0 0 0 R0 = c k1(σ)k2(a)π(a)dσda + . 0 0 [1 − q] (4.4) Here, we note that the threshold parameter R0, usually called the basic reproduction number, and is interpreted as the expected number of secondary cases produced, in a lifetime, by an infective, in a totally susceptible population. In the following result, we determine the steady state solution of problem (2.1) when q =16 . Theorem 4.1 (see El-Doma (2004a) for a proof.) Suppose that q =16 , then: ∗ ∗ (1) If R0 > 1, then λ =0and λ > 0 are possible steady states. A steady state with λ∗ > 0 is unique when it exists and it satisfies ∞ a ∗ a Z Z −λ R k1(τ)dτ 1=c k1(σ)k2(a)π(a)e σ dσda 0 0 ∞ a ∞ ∗ a ∗ a Z Z −λ R k1(τ)dτ Z −λ R k1(τ)dτ cq β(a)π(a)e σ k1(σ)dσda k2(a)π(a)e 0 da 0 0 0 + ∞ . (4.5) Z −λ∗ a k (τ)dτ 1 − q β(a)π(a)e R0 1 da 0 And in this case s∗(a) and i∗(a) are given by ∗ ∗ −λ∗ a k (τ)dτ s (a)=s (0)π(a)e R0 1 , (4.6) a ∗ σ ∗ ∗ ∗ ∗ Z −λ R k1(τ)dτ i (a)=i (0)π(a)+λ s (0)π(a) k1(σ)e 0 dσ, (4.7) 0 where i∗(0) satisfies ∞ a ∗ a ∗ Z Z −λ R k1(τ)dτ cqλ β(a)π(a)k1(σ)e σ dσda ∗ 0 0 i (0) = ∞ . (4.8) Z −λ∗ a k (τ)dτ 1 − q β(a)π(a)e R0 1 da 0 ∗ (2) If R0 ≤ 1, then the disease-free equilibrium, λ =0, is the only steady state, i.e., s∗(a)=cπ(a) and i∗(a)=0. Here, we note that the disease will die out if R0 ≤ 1 and persists if R0 > 1. The effect of vertical transmission via its parameter q is seen, since the right-hand side of (4.4) is an increasing function of q and therefore, a contributing factor for an endemic disease to occur. AAM: Intern. J., Vol. 1, No. 2 (2006) 133 In order to determine the solution of problem (2.1), when q =1, we consider the following transformation, called the age profile of infective: i(a, t) v(a, t)= , p∞(a) and note that s(a, t)=p∞(a) − i(a, t), since we are assuming that the total population has already reached its steady state distribution p∞(a)=cπ(a). Therefore, from (3.6), v(a, t) satisfies the following: ∂v(a, t) ∂v(a, t) + = k1(a)[1 − v(a, t)]λ(t),a>0,t>0, ∂a ∂t ∞ Z v(0,t)=q β(a)π(a)v(a, t)da, t ≥ 0, 0 (4.9) v(a, 0) = v0(a)=i0(a)/p∞(a),a≥ 0, ∞ Z λ(t)=c k2(a)π(a)v(a, t)da, t ≥ 0. 0 Then from (4.9), we see that v(a, t) = 1 is a solution of the above problem when q = 1, and therefore by uniqueness of solution (see section 3), it is the only solution for this case. Therefore, i(a, t)=p∞(a)=cπ(a) is the solution for problem (2.1) when q =1. We note that from (4.7) and q =1, we obtain that ∞ a ∗ σ ∗ ∗ Z Z −λ R k1(τ)dτ λ s (0) β(a)π(a)k1(σ)e 0 dσda =0. (4.10) 0 0 Thus if λ∗ = 0 in (4.10), then using (4.7), we obtain that i∗(0) = 0 (we are assuming ∗ that k2(a) is not identically zero), and therefore i (a)=0. Accordingly, we obtain the following steady state (disease-free equilibrium): s∗(a)=cπ(a),i∗(a)=0. (4.11) ∞ a ∗ σ Z Z −λ R k1(τ)dτ ∗ Now, if we suppose that β(a)π(a)k1(σ)e 0 dσda =0and6 λ =6 0, then 0 0 i∗(0) = c, and thus we obtain the following steady state: ∞ ∗ ∗ ∗ Z i (a)=cπ(a),s (a)=0,λ = c k2(a)π(a)da. (4.12) 0 ∞ a σ Z Z −λ∗R k1(τ)dτ ∗ ∗ Finally, if β(a)π(a)k1(σ)e 0 dσda =0, and λ =06 , then s (0) is 0 0 undetermined and thus for each fixed s∗(0) ∈ [0,c), and using equations (4.7) and (4.3), we obtain ∞ a ∞ σ ∗ ∗ ∗ Z Z − R λ k1(τ)dτ ∗ Z λ 1 − s (0) k2(a)π(a)k1(σ)e 0 dσda =(c − s (0)) k2(a)π(a)da. 0 0 0 (4.13) We note that the left-hand side of (4.13) equals zero when λ∗ =0, and increases to +∞, when λ∗ → +∞. Accordingly, for each fixed s∗(0) ∈ [0,c), we can see that (4.13) gives rise to an endemic equilibrium, and hence problem (2.1) gives rise to a continuum of endemic equilibriums in this special case. 134 Mohammed El-Doma 5 Global Stability Results In this section, we prove the global stability of the endemic equilibriums for problem (2.1). By integrating (4.9) along characteristic lines t − a = const., we find that v(a, t) satisfies t − t k (a−t+τ)λ(τ)dτ Z − t k (a−t+τ)λ(τ)dτ v (a − t)e R0 1 + k (a − t + σ)e Rσ 1 λ(σ)dσ, a > t, 0 1 0 v(a, t)= a − a k (τ)λ(t−a+τ)dτ − a k (τ)λ(t−a+τ)dτ − R0 1 Z − Rσ 1 v(0,t a)e + k1(σ)λ(t a + σ)e dσ, a < t. 0 (5.1) ∞ From (4.9), v(0,t)=qZ β(a)π(a)v(a, t)da, then using (5.1) and changing the order 0 of integration several times and making appropriate changes of variables yields t Z − a k (τ)λ(t−a+τ)dτ v(0,t)=q β(a)π(a)v(0,t− a)e R0 1 da 0 t ∞ a Z Z − R k1(τ)λ(t−a+τ)dτ + β(a)π(a)k1(a − σ)λ(t − σ)e a−σ dadσ (5.2) 0 σ ∞ t Z − R k1(a−t+τ)λ(τ)dτ + β(a)π(a)v0(a − t)e 0 da . t ∞ Z Also, from (4.9), λ(t)=c k2(a)π(a)v(a, t)da, then using (5.1) and changing the 0 order of integration several times and making appropriate changes of variables yields t a Z − R k1(τ)λ(t−a+τ)dτ λ(t)=c k2(a)π(a)v(0,t− a)e 0 da 0 t ∞ a Z Z − R k1(τ)λ(t−a+τ)dτ + k2(a)k1(a − σ)π(a)λ(t − σ)e a−σ dadσ (5.3) 0 σ ∞ t Z − R k1(a−t+τ)λ(τ)dτ + k2(a)π(a)v0(a − t)e 0 da . t Note that by Assumptions 2, 4 and 5 of section 2 and the Dominated Convergence Theorem, we obtain ∞ t Z − R k1(a−t+τ)λ(τ)dτ k2(a)π(a)v0(a − t)e 0 da −→ 0, as t →∞. t Also, by similar reasoning as above, ∞ t Z − R k1(a−t+τ)λ(τ)dτ β(a)π(a)v0(a − t)e 0 da −→ 0, as t →∞. t AAM: Intern. J., Vol. 1, No. 2 (2006) 135 Therefore, setting v(0,t)=u(t), u(t) and λ(t) satisfy the following limiting equations (see Busenberg, et al. (1988)): ∞ Z − a k (τ)λ(t−a+τ)dτ u(t)=q β(a)π(a)u(t − a)e R0 1 da 0 ∞ ∞ a Z Z − R k1(τ)λ(t−a+τ)dτ + β(a)π(a)k1(a − σ)λ(t − σ)e a−σ dadσ , (5.4) 0 σ ∞ a Z − R k1(τ)λ(t−a+τ)dτ λ(t)=c k2(a)π(a)u(t − a)e 0 da 0 ∞ ∞ a Z Z − R k1(τ)λ(t−a+τ)dτ + k2(a)k1(a − σ)π(a)λ(t − σ)e a−σ dadσ . (5.5) 0 σ We integrate equations (5.4 )-(5.5) to obtain the following: ∞ Z − a k (τ)λ(t−a+τ)dτ u(t)=q β(a)π(a)u(t − a)e R0 1 da 0 ∞ Z − a k (τ)λ(t−a+τ)dτ + β(a)π(a) h1 − e R0 1 i da , (5.6) 0 ∞ a Z − R k1(τ)λ(t−a+τ)dτ λ(t)=c k2(a)π(a)u(t − a)e 0 da 0 ∞ a Z − R k1(τ)λ(t−a+τ)dτ + k2(a)π(a) h1 − e 0 i da . (5.7) 0 Now, we set w(t) and g(t) to satisfy the following: w(t)=u(t) − u∗, g(t)=λ(t) − λ∗, i∗(0) where λ∗,u∗ = , are defined as in section 4. Then after some computations, we c obtain that w(t) and g(t) satisfy the following: ∞ Z −λ∗ a k (τ)dτ − a k (τ)g(t−a+τ)dτ w(t)=q β(a)π(a)e R0 1 w(t − a)e R0 1 da 0 ∞ ∗ Z −λ∗ a k (τ)dτ − a k (τ)g(t−a+τ)dτ +[1− u ] β(a)π(a)e R0 1 h1 − e R0 1 i da , (5.8) 0 136 Mohammed El-Doma ∞ ∗ a a Z −λ R k1(τ)dτ − R k1(τ)g(t−a+τ)dτ g(t)=c k2(a)π(a)e 0 w(t − a)e 0 da 0 ∞ ∗ a a ∗ Z −λ R k1(τ)dτ − R k1(τ)g(t−a+τ)dτ +[1− u ] k2(a)π(a)e 0 h1 − e 0 i da . (5.9) 0 In the following theorem, we prove the global stability of the endemic equilibrium when q =16 . Theorem 5.1 If q =16 , then the unique endemic equilibrium, given by Theorem 4.1, is globally stable. Proof. Let |w|∞ = lim sup |w(t)| and |g|∞ = lim sup |g(t)|. Then if we use the fact that t→∞ t→∞ σ σ − R k1(τ)g(t−a+τ)dτ Z 0 1−e 0 ≤ k1(τ)g(t−a+τ)dτ, and then use F atou s Lemma in equations 0 (5.8)-(5.9), we obtain that ∗ ∞ ∞ a q[1 − u ]|g| ∗ a ∞ Z Z −λ R k1(τ)dτ |w| ≤ β(a)π(a)e 0 k1(σ)dσda, (5.10) [1 − q] 0 0 ∞ ∞ a ∗ a ∞ ∞ Z ∗ ∞ Z Z −λ R k1(τ)dτ |g| ≤ c|w| k2(a)π(a)da + c[1 − u ]|g| k2(a)π(a)e 0 k1(σ)dσda. 0 0 0 (5.11) Now, using inequality (5.10) in (5.11), we obtain that ∞ a ∞ q ∗ a ∞ ∞ ∗ Z Z −λ R k1(τ)dτ Z |g| ≤|g| [1 − u ] β(a)π(a)e 0 k1(σ)dσda c k2(a)π(a)da [1 − q] 0 0 0 ∞ a ∗ a Z Z −λ R k1(τ)dτ + c k2(a)π(a)e 0 k1(σ)dσda . (5.12) 0 0 We note that u∗ is given by ∗ [1 − q] 1 − u = ∞ . (5.13) Z −λ∗ a k (τ)dτ 1 − q β(a)π(a)e R0 1 da 0 Also, λ∗ is given by ∞ ∞ ∗ a ∗ Z ∗ Z −λ R k1(τ)dτ λ = c k2(a)π(a)da +[u − 1]c k2(a)π(a))e 0 da. (5.14) 0 0 Now, using equations (5.13)-(5.14) in inequality (5.12) and then using equation (4.5), we obtain |g|∞ < |g|∞ [(1 − u∗)+u∗] . (5.15) AAM: Intern. J., Vol. 1, No. 2 (2006) 137 Accordingly, |g|∞ =0. And hence from inequality (5.10), we obtain that |w|∞ =0. Therefore, the endemic equilibrium is globally stable. In the following theorem, we prove the global stability of the endemic equilibrium when q =1. We note that, if q = 1, then R0 is not defined, and in this case either there exists a unique endemic equilibrium, which is globally stable, or problem (2.1) gives rise to a continuum of endemic equilibriums, in the case of non-fertile infectibles ( i.e., when the support of k1(a) lies to the right of the support of β(a) ), see section 4. ∞ a ∗ σ Z Z −λ R k1(τ)dτ Theorem 5.2 Suppose that q =1, and β(a)π(a)k1(σ)e 0 dσda =06 . 0 0 Then the endemic steady state s∗(a)=0,i∗(a)=cπ(a), is globally stable. Proof. We note that from (5.13), [1 − u∗] → 0asq → 1. Also, from (5.14), we find ∗ → ∞ → that λ c R0 k2(a)π(a)da as q 1. Therefore, using (5.12), we obtain the following as q → 1: ∞ a ∗ a ∗ Z Z −λ R k1(τ)dτ λ β(a)π(a)e 0 k1(σ)dσda ∞ ∞ 0 0 |g| ≤|g| ∞ . (5.16) Z −λ∗ a k (τ)dτ 1 − β(a)π(a)e R0 1 da 0 Now, using equation (4.8), we obtain that |g|∞ = 0, since ∞ a ∗ a ∗ Z Z −λ R k1(τ)dτ λ β(a)π(a)e 0 k1(σ)dσda 0 0 ∞ < 1. Z −λ∗ a k (τ)dτ 1 − β(a)π(a)e R0 1 da 0 Using |g|∞ = 0 in equation (5.10), we obtain that |w|∞ =0. Accordingly, we obtain the global stability for the endemic equilibrium. Conclusion We studied an S I age-structured epidemic model when the disease is vertically as well as horizontally transmitted and the force of infection of proportionate mixing assumption type. The mortality and fertility rates are age-dependent. We note that herpes and AIDS are examples of SI epidemics. We established the well posedness of the model equations and proved global stability results for the endemic equilibriums. If q =16 , then the endemic equilibrium is globally stable. If q =1, then either the endemic equilibrium is the population consisting of infective only, and we proved that this endemic equilibrium is globally stable, or the model gives rise to a continuum of endemic equilibriums, if individuals are susceptible only after the end of their reproductive period. The global stability results that we obtained are under very general conditions, and, in fact, we did not require any condition other than the existence of a unique endemic equilibrium. 138 Mohammed El-Doma Acknowledgments This work is completed while the author is an Arab Regional Fellow at the Center for Ad- vanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, he is supported by a grant from the Arab Fund for Economic and Social Development, and he would like to thank the Director of CAMS, Prof. Dr. Wafic Sabra, for an invitation and hospitality during his stay in CAMS. The author would also like to thank two anonymous referees for pointing out typo- graphical and grammetical errors. References [1] Anderson, R. M. and R. M. May, Infectious diseases of humans, Dynamic and control, Oxford University Press, (1999). [2] Bellman, R. and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, (1963). [3] Brauer, F., A model for an SI disease in an age-structured population. Discrete and Continuous Dynamical Systems-Series B. Vol. 2, pp. 257-264, (2002a). [4] Brauer, F. Infectious disease models with chronological age structure and epidemi- ological age structure, In: Mathematical Approach of Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Carlos Castillo-Chavez with Sally Blower, Pauline van den Driessche, Densie Kirshner, and Abdul-Aziz Yakubu, (Eds), IMA Vol. 126, pp. 231-243, Springer-Verlag, Berlin, (2002b). [5] Busenberg, S. N. and K.L. Cooke, Vertically transmitted diseases. Models and dy- namics, Biomathematics, Vol. 23, Springer-Verlag, Berlin, (1993). [6] Busenberg, S. N., K. L. Cooke and M. Iannelli. Endemic thresholds and stability in a class of age-structured epidemics. SIAM J. Appl. Math. Vol. 48, No. 6, pp. 1379- 1395, (1988). [7] Busenberg, S. N., and C. Casttillo-Chavez. A general sloution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS. IMA J. Math. Appl. Med. & Biol. Vol. 8, pp. 1-29, (1991). [8] Dietz, K., and D. Schenzle. Proportionate mixing models for age dependent infection transmission. J. Math. Biol. Vol. 22, pp. 117-120, (1985). [9] El-Doma, M., Analysis of nonlinear integro-differential equations arising in age- dependent epidemic models. Nonlinear Analysis TMA. Vol. 11, pp. 913-937, (1987). [10] El-Doma, M., A global stability result and existence and uniqueness of an age-dependent SI epidemic with disease-induced mortality and propor- tionate mixing assumption: The case of vertically transmitted diseases. International Journal of Ecology & Development. Vol. 4, pp. 52-65, ( 2006). AAM: Intern. J., Vol. 1, No. 2 (2006) 139 [11] El-Doma, M. The Existence and Uniqueness of Solution to an Age-structured SI Epi- demic Model with Disease-Induced Mortality and Vertical Transmission, Proceedings of the International Conference on Mathematics and its Applications, April 5-7, 2004, Department of Mathematics & Computer Science, Kuwait University, Kuwait, pp. 126-129, (2004c). [12] El-Doma, M., Analysis of an age-dependent SI epidemic model with disease-induced mortality and proportionate mixing assumption: The case of vertically transmitted diseases. Journal of Applied mathematics. Vol. 3, pp. 235-253, (2004b). [13] El-Doma, M., Analysis of an SI age-structured epidemic model with vertical transmission and proportionate mixing assumption. Mathematical Sciences Research Journal. Vol. 8, pp. 239-260, (2004a). [14] El-Doma, M., Analysis of an age-dependent SI epidemic model with disease-induced mortality and proportionate mixing assumption. International Journal of Applied Mathematics. Vol. 3, pp. 233-247, (2000). [15] Feller, W., On the integral equation of renewal theory. Ann. Math. Stat. Vol. 12, pp. 243-267, (1941). [16] Hoppensteadt, F. Mathematical theory of population demographics, genetics and epidemics, CBMS-NSF Regional Conference in Applied Mathematics, Philadelphia, (1975). [17] Louie, K., M. G. Roberts and G. C. Wake. The regulation of an age-structured population by a fatal disease. IMA J. Math. Appl. Med. Biol. Vol. 11, pp. 229-244, (1994). [18] May, R. M., R. M. Anderson and A. R. McLean. Possible demographic conse- quences of HIV/AIDS epidemics. I. Assuming HIV infection always leads to AIDS. Math. Biosci. Vol. 90, pp. 475-505, (1988).